Consider the vector field a(r) = re+ (CT) Show that a is irrotational Find a scalar...
10. For each of the following domains either explain why every irrotational vector field is conservative or give (and verify) a nonconservative irrotational example (i) R i) R3 (0,0,0 R3 r-y 0)
10. For each of the following domains either explain why every irrotational vector field is conservative or give (and verify) a nonconservative irrotational example (i) R i) R3 (0,0,0 R3 r-y 0)
Please make it simple and
clear to understand
3. A vector field is given by (a) Show that the vector field r is conservative. Then find a scalar potential function f(r,y,) such that r - gradf and f(0,0,0) 0 (b) By the result of (a) the following line integral is path independent. Using the scalar potential obtained in (a) evaluate the integral from (0,0,2) (where-y-0) to (4,2,3) (where -1,y 0,2) 4.2,3) J(0,0,2)
3. A vector field is given by (a)...
a) What does the solenoidal vector field and irrotational vector field mean, what does it mean physically? Show that in a single mathematical expression, a vector field A is solenoidal and irrotational, respectively. b) A solenoidal field vector along the surface integral of a closed surface is equal to 0 to show through the divergence theorem. c) Show by means of the Stokes theorem that the line integral of an irrotational vector field along the closed curve surrounding a surface...
We can combine the scalar potential V and the vector potential A
to a combined 4-vector potential:
Calculate the components of a 4x4 electromagnetic field
tensor:
with the contravariant vector:
from the electric field
and the magnetic field
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2. Consider a static volume current density J(r') where r' is the position vector of a point in the current distribution. Show that the field generated at a point with position vector r, according to the Biot-Savart law, u mrJ(r')RJ B(r) = -JJJp3 av, 477 o in which R=r-r' and R=R , satisfies Maxwell's magnetostatic equation V x B = 4J (u should be considered as constant). Consider the magnetic vector potential defined by A and the Lorenz gauge Show...
Q.8-) Given the vector field; (a) Is B irrotational(or conservative)? (b) Find the net flux of B over the cube Ocx,y,z1. (c) Determine the circulation of Baround the edge of the square z-o,ox,ysi. Assume anticlockwise direction.
4. Consider the vector field u = (3r+yz) region V bounded by 2y2 < (2 - z)2 for y 2 0 and 0 y)j+(xy+2z)k, defined across a three-dimensional 1. z (a) Show that u is conservative and find a scalar function d that satisfies u = Vo. [6 marks] (b) Sketch the volume V and express the limits of the volume V in terms of cylindrical coordi nates (r, 0, z) [3 marks (c) Using the divergence theorem calculate the...
Please answer without using previously posted answers.
Thanks
Let F(x, y) be a two-dimensional vector field. Spose further that there exists a scalar function, o, such that Then, F(x,y) is called a gradient field, and φ s called a potential function. Ideal Fluid Flow Let F represent the two-dimensional velocity field of an inviscid fluid that is incompressible, ie. . F-0 (or divergence-free). F can be represented by (1), where ф is called the velocity potential-show that o is harmonic....
Let F = (2,1,1) be a vector field in Rº a.) Show that F is a conservative vector field. b.) Find the potential function of F. In other words, find a scalar function f(x, y, z) such that ✓ f = 7. Please show all steps. c.) Let C be any smooth curve starting at (1,1,1) and ending at (e, e, 1). Compute (Fdi. С
Could you please answer both and show work?
1.13.10 With E the electric field and A the magnetic vector potential, show that [E + aA/ai] is irrotational and that therefore we may write 0t 1.13.11 The total force on a charge q moving with velocity v is Using the scalar and vector potentials, show that Exercise 1.13.10